Quasi-periodic fast-propagating (QFP) magnetosonic waves and extreme ultraviolet (EUV) waves were proposed to be driven by solar flares and coronal mass ejections (CMEs), respectively. In this Letter, we present a detailed analysis of an interesting event in which we find that both QFP magnetosonic waves and EUV waves are excited simultaneously in one solar eruption event. The co-existence of the two wave phenomena offers an excellent opportunity to explore their driving mechanisms. The QFP waves propagate in a funnel-like loop system with a speed of 682--837 speed{} and a lifetime of 2 minutes. On the contrary, the EUV waves, which present a faster component and a slower component, propagate in a wide angular extent, experiencing reflection and refraction across a magnetic quasi-separatrix layer. The faster component of the EUV waves travels with a speed of 412--1287 speed{}, whereas the slower component travels with a speed of 246--390 speed{}. The lifetime of the EUV waves is $sim$15 minutes. It is revealed that the faster component of the EUV waves is cospatial with the first wavefront of the QFP wave train. Besides, The QFP waves have a period of about 45$pm$5 seconds, which is absent in the associated flares. All these results imply that QFP waves can also be excited by mass ejections, including CMEs or jets.
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